Tze Chun Tsang, aged 51, was the front seat passenger in a black Peugeot which failed to stop for officers.
Attempts were made to get the car to pull over in Brierley, Barnsley, but the motorist behind the wheel refused to stop.
A police pursuit was mounted and the car was followed towards the junction of Park Road and Brierley Road, where it was involved in a collision with a silver Ford.
The driver of the Peugeot and two other occupants fled, leaving Mr Tsang critically injured.
It has not yet been revealed whether they have been tracked down.
The occupants of the Ford escaped with minor injuries.
Paying tribute to Mr Tsang, his family said: “It is with deep sadness we announce the death of a much loved son, dad and brother.
“Father of three, Tze Chun Tsang, died in hospital on Saturday, May 8 during the early hours.”
Witnesses to the collision or motorists in the area at the time who have dashcams should come forward.
South Yorkshire Police said: “At 5.55pm on Sunday, May 2, police attempted to pull over a car in the Brierley area of Barnsley when it failed to stop for officers.
“The car, a black Peugeot, was pursued by police towards Park Road.
“At the junction of Park Road and Brierley Road, the Peugeot was involved in a collision with a second car, a silver Ford.
“Enquiries are ongoing and officers are asking anyone with information to come forward.”